arXiv research

The paper introduces surfaces with constant solid angle for designing shell structures.

We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties of the solid angle maximizer. The other is the pair of a planar compact subset an…

This note generalizes the visual angle to convex sets in 3D space.

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is …

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $Φ\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $Φ$ is an explicit Seifert hypersurface for $M$.

Researchers solved a geometry paradox for creased tubes.

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must …

We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid tori with an arbitrary conical singularity along the core. Our method is to decom…

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

New periodic polyhedra found in curved spaces.

In this paper we classify certain special ruled surfaces in $\R^3$ under the general theorem of characterization of constant angle surfaces. We study the tangent developable and conical surfaces from the point of view the constant angle property. Moreover, the natural extension to normal and binormal constant angle sur…

Study on null hypersurfaces with constant angle in Lorentzian manifolds.

We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…

We study surfaces in $\R^4$ whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We …

A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder…

In the present paper we classify curves and surfaces in Euclidean $3-$space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini's surface in terms of constant angle surfaces.

Minimal surfaces span periodic curves in 3D space.

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.

The paper classifies hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with constant curvature.

In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction.

The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.

Study on metrics on manifolds with specific curvature properties.

In this paper we classify constant angle surfaces in $\H^2\times\R$, where $\H^2$ is the hyperbolic plane.

Given a warped product of the real line with a Riemannian manifold of arbitrary dimension, we classify the hypersurfaces whose tangent spaces make a constant angle with the vector field tangent to the real direction. We show that this is a natural setting in which to extend previous results in this direction made by se…

In this paper we classify all surfaces in the 3-dimensional Lie group $Sol_3$ whose normals make constant angle with a left invariant vector field.

To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Laws…

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.

In this paper we introduce the notion of contact angle for an immersed surface in three dimensional sphere. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in $S^3$ with constant contact angle. Also, we give an example of a minimal surface in $S^3$ with non constant…

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…

We study Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$ with respect to their, so called, angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follo…

In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.

Classifies isoparametric hypersurfaces in 3D manifolds.

Paper explores folding patterns of curved creases preserving their geometric properties.

Study classifies helix surfaces in Lorentzian Heisenberg group.

The study classifies isoparametric hypersurfaces in product spaces with constant angle function.

In this article we generalize the notion of constant angle surfaces in S^2 x R and H^2 x R to general Bianchi-Cartan-Vranceanu spaces, i.e. essentially to three-dimensional homogeneous spaces with a four-dimensional isometry group. We show that these surfaces have constant Gaussian curvature and we give a complete loca…

In this article we study surfaces in $\mathbb{S}^3(1) \times \mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector.

Minimal covolume group found in hyperbolic 3-space.

Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.

The study classifies isoparametric hypersurfaces in specific product spaces.

The paper explores connections between perimeter, area, and visual angle of convex sets.

The paper classifies various types of hypersurfaces in a product space.

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criteri…

We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.